Msm Analysis

2. Build Count Matrix

from scipy import linalg

C = np.zeros((n_states, n_states))
for q in range(n_traj):
    for t in range(n_frames - tau):
        C[labels[q, t], labels[q, t + tau]] += 1

3. Transition Matrix + Implied Timescales

row_sums = C.sum(axis=1, keepdims=True)
row_sums[row_sums == 0] = 1
T = C / row_sums

evals = np.sort(np.abs(np.real(linalg.eigvals(T))))[::-1]
dt = tau * timestep_ps  # lag time in ps
timescales = [-dt / np.log(ev) for ev in evals[1:] if 0 < ev < 1]

4. Lag Time Scan

Scan τ from 1 to K/4. Implied timescales should plateau — that's your converged estimate.

OOM Bias Correction (Nüske et al. Eqs. 44-55)

Algorithm

1. Count vector s: initial state frequencies
2. Count matrix S^τ: pair counts at lag τ
3. Two-step tensor S^{2τ}: triplet counts at lags τ and 2τ
4. SVD of S^τ → truncate to rank M
5. Project into M-dimensional space → eigenvalues give corrected timescales

Key Issue: Sparsity

The two-step tensor S^{2τ} has n_states³ bins. With 40 states = 64K bins.

• Need >> 64K total triplet counts for reliable OOM
• Rule of thumb: total_counts / n_states³ > 50 for each nonzero bin
• If sparse: reduce states or use longer trajectories

Practical Results

Lesson: OOM corrects in the right direction but overshoots with sparse data. Direct MSM with sufficient data and appropriate τ converges to the correct timescale.

Alanine Dipeptide Specifics

Starting Structure Requirements

• MUST sample both metastable basins (αR: φ<0, αL: φ>0)
• Short equilibration from single conformation → never crosses barrier → biased data
• Fix: High-temperature exploration (600-1000K) to find both basins, then cool to 300K

Trajectory Length

• Paper: K=5000 frames per trajectory
• Minimum for OOM: K≥2000 frames (100+ ps at 50fs save interval)
• K=401 frames (20ps): Direct MSM works, OOM too sparse

Simulation Parameters

# OpenMM setup
integrator = mm.LangevinMiddleIntegrator(300*unit.kelvin, 1/unit.picosecond, 2*unit.femtoseconds)
# Save every 25 steps (50 fs)
# 2000 saves = 100 ps per trajectory